De Turck trick on mean curvature flow

Question

I am reading the book Lecture on mean curvature flow by Xi-Ping Zhu.

Suppose $M^n$ is an n-dimension smooth manifold and $X(x,t):M^n \rightarrow R^{n+1}$ be a one-parameter family of smooth immersion. Metric and the second fundamental form on $X(x,t)$ is defined as $$g_{ij} = (\frac{\partial X }{\partial x_i},\frac{\partial X }{\partial x_j})\,\,\,\,h_{ij}= (n(x,t),\frac{\partial^2 X }{\partial x_i\partial x_j})$$ And the covariant derivative of vector $v$ is defined by $$\nabla_j v_i = \frac{\partial v_i }{\partial x_j} + \Gamma^i_{jk}v_k$$ By setting $$\frac{\partial X }{\partial t}=H n$$ Using the Gauss equation and the Weingarten equation, the author further calculate that $$g^{ij}\nabla_i\nabla_j X=g^{ij}(\frac{\partial^2 X }{\partial x_i\partial x_j}-\Gamma_{ij}^k\frac{\partial X }{\partial x_k})=g^{ij}h_{ij}n=Hn=\frac{\partial X }{\partial t}$$ Later he used the De Turck trick to make this equation become strictly parabolic. But I think $$\frac{\partial X }{\partial t}=g^{ij}(\frac{\partial^2 X }{\partial x_i\partial x_j}-\Gamma_{ij}^k\frac{\partial X }{\partial x_k})$$ is already strictly parabolic. what is the purpose of the trick? Thank you so much.

Answer 1

Note that if $p\in M$ the symbol on some $\xi$ direction is given by $\sigma(\xi,p) = g^{ij}\xi_i\xi_j - g^{ij}\Gamma_{ij}^k\xi_k = |\xi|^2_{g} - g^{ij}\Gamma_{ij}^k\xi_k,$ and if you choose $g^{ij}\Gamma_{ij}^k\xi_k = |\xi|_g^2,$ the symbol is not an isomorphism.